Question

Separable differential equations: Suppose a population P = P(t) grows in such a way that its rate of growth $\frac{dP}{dt}$obeys the equation$\frac{dP}{dt}=P\left(100-P\right)$

This is called a differential equation because it is an equation that involves a derivative. In the series of steps that follow, you will find a function P(t) that behaves according to this differential equation.

As we will see in Section 6.5, a differential equation of this form is separable to the following equivalent equation involving integrals:

$\int \frac{dP}{P\left(100-P\right)}=\int 1dt$

Short answer

The given statement is proved.

Step-by-step solution

Step 1. Given information

Differential equation is$\frac{dP}{dt}=P\left(100-P\right)$

Step 2. Explanation

$\frac{dP}{dt}=P\left(100-P\right)$

Integrating both sides.

$\int \frac{dP}{P\left(100-P\right)}=\int 1dt$